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r
T
DC
1
c
+d
s
+d
T
0
0
+ d
0
Figure 11.4b Stress states in the Mohr-Coulomb diagram
Consider an elementary curved volume in this system with normal and shear
stresses and their increments, shown in Fig 11.4a. The effect of shear increments
on surface BD and CD is covered by the shear stress working on ED, and
components of this residual shear stress (
2
) are accounted for in the
s-
and
r-
direction. Then, equilibrium yields (note,
,s
expresses
d
/ds
, etc.)
0,
s
dsdr + 2
,s
dsdr +
dsdr
sin(
)/cos
=
0
0
,r
dsdr
2
,r
dsdr +
dsdr
cos
/cos
=
0
Introducing (11.3) yields the equations of Kötter
0,
s
+
2
0
tan
,s
+
sin(
)/cos
=
0
along direction
s
(11.4a)
0,
r
2
0
tan
,r
+
cos
/
cos
=
0
along direction
r
(11.4b)
Disregarding the specific weight (
=
0) equations (11.4) become for a cohesive
soil (
=
0
,
c = c
u
)
,s
+
2
c
u
,s
=
0
or
+
2
c
u
= C
s
along direction
s
(11.5a)
,r
2
c
u
,r
=
0
or
2
c
u
= C
r
along direction
r
(11.5b)
These equations form a set of hyperbolic partial differential equations. Their
solution contains so-called characteristic lines, i.e. slip lines, with changing angle
and rotating principal stresses. The same holds for drained soils (
0), graphically
shown in Fig 11.4a and Fig 11.4b.
In some cases, e.g. when there is no friction at the border of a failing zone
(Rankine stress, Chapter 8), the slip lines are straight. Soil-wall friction (at the
border) causes curving slip lines; some examples are shown in Fig 11.5.
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